Graphical Approach to College Algebra, A

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  • Edition: 4th
  • Format: Hardcover
  • Copyright: 2007-01-01
  • Publisher: Addison Wesley
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This edition has evolved to address the needs of today's student. While maintaining its unique table of contents and functions-based approach, the text now includes additional components to build skill, address critical thinking, solve applications, and apply technology to support traditional algebraic solutions. It continues to incorporate an open design, helpful features, careful explanations of topics, and a comprehensive package of supplements and study aids to provide new and relevant opportunities for learning and teaching.

Table of Contents

Prefacep. xii
Linear Functions, Equations, and Inequalitiesp. 1
Real Numbers and the Rectangular Coordinate Systemp. 2
Sets of Real Numbers
The Rectangular Coordinate System
Viewing Windows
Distance and Midpoint Formulas
Introduction to Relations and Functionsp. 12
Set-Builder Notation and Interval Notation
Relations, Domain, and Range
Function Notation
Reviewing Basic Concepts (Sections 1.1 and 1.2)p. 22
Linear Functionsp. 23
Basic Concepts about Linear Functions
Slope of a Line
Slope-Intercept Form of the Equation of a Line
Equations of Lines and Linear Modelsp. 36
Point-Slope Form of the Equation of a Line
Standard Form of the Equation of a Line
Parallel and Perpendicular Lines
Linear Models and Regression
Reviewing Basic Concepts (Sections 1.3 and 1.4)p. 50
Linear Equations and Inequalitiesp. 51
Solving Linear Equations
Graphical Approaches to Solving Linear Equations
Identities and Contradictions
Solving Linear Inequalities
Graphical Approaches to Solving Linear Inequalities
Three-Part Inequalities
Applications of Linear Functionsp. 66
Problem-Solving Strategies
Applications of Linear Equations
Break-Even Analysis
Direct Variation
Reviewing Basic Concepts (Sections 1.5 and 1.6)p. 78
Chapter 1 Summaryp. 79
Chapter 1 Review Exercisesp. 82
Chapter 1 Testp. 87
Chapter 1 Project Predicting Heights and Weights of Athletesp. 88
Analysis of Graphs of Functionsp. 89
Graphs of Basic Functions and Relations; Symmetryp. 90
Increasing and Decreasing Functions
The Identity Function
The Squaring Function and Symmetry with Respect to the y-Axis
The Cubing Function and Symmetry with Respect to the Origin
The Square Root and Cube Root Functions
The Absolute Value Function
The Relation x = y[superscript 2] and Symmetry with Respect to the x-Axis
Even and Odd Functions
Vertical and Horizontal Shifts of Graphsp. 103
Vertical Shifts
Horizontal Shifts
Combinations of Vertical and Horizontal Shifts
Effects of Shifts on Domain and Range
Horizontal Shifts Applied to Equations for Modeling
Stretching, Shrinking, and Reflecting Graphsp. 113
Vertical Stretching
Vertical Shrinking
Horizontal Stretching and Shrinking
Reflecting across an Axis
Combining Transformations of Graphs
Reviewing Basic Concepts (Sections 2.1-2.3)p. 125
Absolute Value Functions: Graphs, Equations, Inequalities, and Applicationsp. 127
The Graph of y = [vertical bar]f(x)[vertical bar]
Properties of Absolute Value
Equations and Inequalities Involving Absolute Value
An Application Involving Absolute Value
Piecewise-Defined Functionsp. 138
Graphing Piecewise-Defined Functions
The Greatest Integer Function
Applications of Piecewise-Defined Functions
Operations and Compositionp. 149
Operations on Functions
The Difference Quotient
Composition of Functions
Applications of Operations and Composition
Reviewing Basic Concepts (Sections 2.4-2.6)p. 162
Chapter 2 Summaryp. 163
Chapter 2 Review Exercisesp. 166
Chapter 2 Testp. 169
Chapter 2 Project Modeling the Movement of a Cold Frontp. 171
Polynomial Functionsp. 173
Complex Numbersp. 174
The Number i
Operations with Complex Numbers
Quadratic Functions and Graphsp. 181
Completing the Square
Graphs of Quadratic Functions
Vertex Formula
Extreme Values
Applications and Quadratic Models
Quadratic Equations and Inequalitiesp. 194
Zero-Product Property
Solving x[superscript 2] = k
Quadratic Formula and the Discriminant
Solving Quadratic Equations
Solving Quadratic Inequalities
Formulas Involving Quadratics
Another Quadratic Model
Reviewing Basic Concepts (Sections 3.1-3.3)p. 208
Further Applications of Quadratic Functions and Modelsp. 208
Applications of Quadratic Functions
Quadratic Models
Higher-Degree Polynomial Functions and Graphsp. 218
Cubic Functions
Quartic Functions
End Behavior
x-Intercepts (Real Zeros)
Comprehensive Graphs
Curve Fitting and Polynomial Models
Reviewing Basic Concepts (Sections 3.4 and 3.5)p. 231
Topics in the Theory of Polynomial Functions (I)p. 231
Intermediate Value Theorem
Division of Polynomials and Synthetic Division
Remainder and Factor Theorems
Topics in the Theory of Polynomial Functions (II)p. 240
Complex Zeros and the Fundamental Theorem of Algebra
Number of Zeros
Rational Zeros Theorem
Descartes' Rule of Signs
Boundedness Theorem
Polynomial Equations and Inequalities; Further Applications and Modelsp. 251
Polynomial Equations and Inequalities
Complex nth Roots
Applications and Polynomial Models
Reviewing Basic Concepts (Sections 3.6-3.8)p. 259
Chapter 3 Summaryp. 260
Chapter 3 Review Exercisesp. 263
Chapter 3 Testp. 267
Chapter 3 Project Creating a Social Security Polynomialp. 268
Rational, Power, and Root Functionsp. 270
Rational Functions and Graphsp. 271
The Reciprocal Function
The Rational Function Defined by f(x) = 1/x[superscript 2]
More on Graphs of Rational Functionsp. 277
Vertical and Horizontal Asymptotes
Graphing Techniques
Oblique Asymptotes
Graphs with Points of Discontinuity
Rational Equations, Inequalities, Applications, and Modelsp. 291
Solving Rational Equations and Inequalities
Applications and Models of Rational Functions
Inverse Variation
Combined and Joint Variation
Reviewing Basic Concepts (Sections 4.1-4.3)p. 305
Functions Defined by Powers and Rootsp. 306
Power and Root Functions
Modeling Using Power Functions
Graphs of f(x) = [characters not reproducible]
Graphing Circles and Horizontal Parabolas Using Root Functions
Equations, Inequalities, and Applications Involving Root Functionsp. 318
Equations and Inequalities
An Application of Root Functions
Reviewing Basic Concepts (Sections 4.4 and 4.5)p. 328
Chapter 4 Summaryp. 329
Chapter 4 Review Exercisesp. 331
Chapter 4 Testp. 335
Chapter 4 Project How Rugged Is Your Coastline?p. 336
Inverse, Exponential, and Logarithmic Functionsp. 338
Inverse Functionsp. 339
Inverse Operations
One-to-One Functions
Inverse Functions and Their Graphs
Equations of Inverse Functions
An Application of Inverse Functions
Exponential Functionsp. 350
Real-Number Exponents
Graphs of Exponential Functions
Exponential Equations (Type 1)
Compound Interest
The Number e and Continuous Compounding
An Application of Exponential Functions
Logarithms and Their Propertiesp. 363
Definition of Logarithm
Common Logarithms
Natural Logarithms
Properties of Logarithms
Change-of-Base Rule
Reviewing Basic Concepts (Sections 5.1-5.3)p. 373
Logarithmic Functionsp. 374
Graphs of Logarithmic Functions
Applying Earlier Work to Logarithmic Functions
A Logarithmic Model
Exponential and Logarithmic Equations and Inequalitiesp. 384
Exponential Equations and Inequalities (Type 2)
Logarithmic Equations and Inequalities
Equations and Inequalities Involving Both Exponentials and Logarithms
Formulas Involving Exponentials and Logarithms
Reviewing Basic Concepts (Sections 5.4 and 5.5)p. 393
Further Applications and Modeling with Exponential and Logarithmic Functionsp. 394
Physical Science Applications
Financial Applications
Biological and Medical Applications
Modeling Data with Exponential and Logarithmic Functions
Chapter 5 Summaryp. 408
Chapter 5 Review Exercisesp. 411
Chapter 5 Testp. 414
Chapter 5 Project Modeling Motor Vehicle Sales in the United States (with a lesson about the careless use of mathematical models)p. 415
Analytic Geometryp. 417
Circles and Parabolasp. 418
Conic Sections
Equations and Graphs of Circles
Equations and Graphs of Parabolas
Translations of Parabolas
An Application of Parabolas
Ellipses and Hyperbolasp. 432
Equations and Graphs of Ellipses
Translations of Ellipses
An Application of Ellipses
Equations and Graphs of Hyperbolas
Translations of Hyperbolas
Reviewing Basic Concepts (Sections 6.1 and 6.2)p. 445
Summary of the Conic Sectionsp. 445
Identifying Conic Sections
Parametric Equationsp. 454
Graphs of Parametric Equations and Their Rectangular Equivalents
Alternative Forms of Parametric Equations
An Application of Parametric Equations
Reviewing Basic Concepts (Sections 6.3 and 6.4)p. 458
Chapter 6 Summaryp. 459
Chapter 6 Review Exercisesp. 461
Chapter 6 Testp. 463
Chapter 6 Project Modeling the Path of a Bouncing Ballp. 464
Systems of Equations and Inequalities; Matricesp. 466
Systems of Equationsp. 467
Linear Systems
Substitution Method
Elimination Method
Special Systems
Nonlinear Systems
Applications of Systems
Solution of Linear Systems in Three Variablesp. 480
Geometric Considerations
Analytic Solution of Systems in Three Variables
Applications of Systems
Curve Fitting Using a System
Solution of Linear Systems by Row Transformationsp. 488
Matrix Row Transformations
Row Echelon Method
Reduced Row Echelon Method
Special Cases
An Application of Matrices
Reviewing Basic Concepts (Sections 7.1-7.3)p. 499
Matrix Properties and Operationsp. 500
Terminology of Matrices
Operations on Matrices
Applying Matrix Algebra
Determinants and Cramer's Rulep. 513
Determinants of 2 x 2 Matrices
Determinants of Larger Matrices
Derivation of Cramer's Rule
Using Cramer's Rule to Solve Systems
Solution of Linear Systems by Matrix Inversesp. 524
Identity Matrices
Multiplicative Inverses of Square Matrices
Using Determinants to Find Inverses
Solving Linear Systems Using Inverse Matrices
Curve Fitting Using a System
Reviewing Basic Concepts (Sections 7.4-7.6)p. 536
Systems of Inequalities and Linear Programmingp. 537
Solving Linear Inequalities
Solving Systems of Inequalities
Linear Programming
Partial Fractionsp. 547
Decomposition of Rational Expressions
Distinct Linear Factors
Repeated Linear Factors
Distinct Linear and Quadratic Factors
Repeated Quadratic Factors
Reviewing Basic Concepts (Sections 7.7 and 7.8)p. 554
Chapter 7 Summaryp. 554
Chapter 7 Review Exercisesp. 557
Chapter 7 Testp. 561
Chapter 7 Project Finding a Polynomial Whose Graph Passes through Any Number of Given Pointsp. 562
Further Topics in Algebrap. 565
Sequences and Seriesp. 566
Series and Summation Notation
Summation Properties
Arithmetic Sequences and Seriesp. 576
Arithmetic Sequences
Arithmetic Series
Geometric Sequences and Seriesp. 584
Geometric Sequences
Geometric Series
Infinite Geometric Series
Reviewing Basic Concepts (Sections 8.1-8.3)p. 594
The Binomial Theoremp. 595
A Binomial Expansion Pattern
Pascal's Triangle
Binomial Coefficients
The Binomial Theorem
rth Term of a Binomial Expansion
Mathematical Inductionp. 602
Proof by Mathematical Induction
Proving Statements
Generalized Principle of Mathematical Induction
Proof of the Binomial Theorem
Reviewing Basic Concepts (Sections 8.4 and 8.5)p. 608
Counting Theoryp. 608
Fundamental Principle of Counting
Distinguishing between Permutations and Combinations
Probabilityp. 617
Basic Concepts
Complements and Venn Diagrams
Union of Two Events
Binomial Probability
Reviewing Basic Concepts (Sections 8.6 and 8.7)p. 626
Chapter 8 Summaryp. 627
Chapter 8 Review Exercisesp. 631
Chapter 8 Testp. 633
Chapter 8 Project Using Experimental Probabilities to Simulate Family Makeupp. 634
Reference: Basic Algebraic Conceptsp. 637
Review of Exponents and Polynomialsp. 638
Rules for Exponents
Terminology for Polynomials
Adding and Subtracting Polynomials
Multiplying Polynomials
Review of Factoringp. 644
Factoring Out the Greatest Common Factor
Factoring by Grouping
Factoring Trinomials
Factoring Special Products
Factoring by Substitution
Review of Rational Expressionsp. 651
Domain of a Rational Expression
Lowest Terms of a Rational Expression
Multiplying and Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Complex Fractions
Review of Negative and Rational Exponentsp. 659
Negative Exponents and the Quotient Rule
Rational Exponents
Review of Radicalsp. 665
Radical Notation
Rules for Radicals
Simplifying Radicals
Operations with Radicals
Rationalizing Denominators
Chapter R Testp. 673
Geometry Formulasp. 674
Deciding Which Model Best Fits a Set of Datap. 676
Answers to Selected Exercisesp. 1
Index of Applicationsp. 1
Indexp. 5
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